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numerical analysis

Questions and Answers of Numerical Analysis

Let f: X → Y be a linear homeomorphism (remark 2.12). Then there exists constants m and M such that for all x1, x2 ∊ X, m||x1 - x2|| ≤ || f (x1) - f (x2)|| ≤ M||x1 - x2||
Let X and Y be Banach spaces. Any linear function f: X → Y is continuous if and only if its graph graph (f) = {(x, y) : y = f(x), x ∊ X} is a closed subset of X × Y.
Show that f (x) 2x + 3 violates both the additivity and the homogeneity requirements of linearity.
A function f: X → Y is affine if and only if f(x) = g(x) + y where g: X → Y is linear and y ∊ Y
Show that the function f: ℜ3 → 3 ℜ2 defined by f(x1, x2, x3) = (x1 = x2, 0) is a linear function. Describe this mapping geometrically.
Show that an affine function maps affine sets to affine sets, and vice versa. That is, if S is an affine subset of X, then f (S) is an affine subset of Y; if T is an affine subset of Y, then f-1 (T)
An affine function preserves convexity; that is, S ⊆ X convex implies that f (S) is convex.
If the production plan y ∊ Y maximizes profits at prices p > 0, then y is efficient (example 1.61).
Assume that the sample space S is finite. Then the expectation functional E takes the formwith ps ‰¥ 0 and ˆ‘sˆŠS ps = 1. ps = P({s}) is the probability of state s.
Let X = C[0; 1] be the space of all continuous functions x(t) on the interval 0, 1. Show that the functional defined by f(x) = x(1/2) is a linear functional on C[0; 1].
Let {S1, S2, Sn} be a collection of subsets of a linear space X with S = S1 + S2 + ... + Sn. Let f be a linear functional on X. Then x* = x1* + x*2 + ... + x*n maximizes f over S if and only if x*i
Let C0 denote the subspace of lm consisting of all infinite sequences converging to zero, that is C0 = {(xt) ∊ l∞: xt → 0}. Show that 1. l1 ⊂ c0 ⊂ l∞ 2. l∞ is the dual of C0 3. l∞ is
Let X be a linear space and
Let f, g1, g2, ..., gm be linear functionals on a linear space X. f is linearly dependent on g1, g2, ..., gm, that is, f ˆŠ lin g1, g2, ..., gm if and only if
H is a hyperplane in a linear space X if and only if there exists a nonzero linear functional f ∊ X' such that H = {x ∊ X : f(x) = c} for some c ∊ ℜ. We use Hf (c) to denote the specific
Describe the action of the mapping f : „œ2 †’ „œ2 defined by
Let H be a hyperplane in a linear space that is not a subspace. Then there is a unique linear functional f ∊ X' such that H = {x ∊ X: f (x) = 1} On the other hand, where H is a subspace, we have
Let H be a maximal proper subspace of a linear space X and x0 ˆ‰ H. (H is a hyperplane containing 0). There exists a unique linear functional f ˆŠ X€².Such that H = {x
For any f, g ∊ Xʹ Kernel f = kernel g ⇔ f =
Let f be a nonzero linear functional on a normed linear space X. The hyperplaneH = {x ∊ X: f(x) = c}is closed if and only if f is continuous.
Show that the function defined in the previous example is bilinear. There is an intimate relationship between bilinear functionals and matrices, paralleling the relationship between linear functions
Let f: X Ã— Y †’ „œ be a bilinear functional on finite-dimensional linear spaces X and Y. Let m = dim X and n = dim Y. For every choice of bases for X and Y, there
Show that the function f defined in the preceding example is bilinear.
Let BiL(X × Y, Z) denote the set of all continuous bilinear functions from X × Y to Z. Show that BiL(X × Y, Z) is a linear space. The following result may seem rather esoteric but is really a
Let X, Y, Z be linear spaces. The set BL(Y, Z) of all bounded linear functions from Y to Z is a linear space (exercise 3.33). Let BL(X, BL(Y, Z)) denote the set of bounded linear functions from X to
Every symmetric, nonnegative definite bilinear functional f satisfies the inequality (f (x, y))2 ≤ f (x, x)f (y, y) for every x, y ∊ X. A symmetric, positive definite bilinear functional on a
Show that the Shapley value ϕ defined by (1) is linear.
For every x, y in an inner product space, |xT y| ≤ ||x|| ||y||
The inner product is a continuous bilinear functional.
The functional ||x|| = √xTx is a norm on X.
Every element y in an inner product space X defines a continuous linear functional on X by fy(x) = xT y.
A nonempty compact convex set in an inner product space has at least one extreme point.
In an inner product space ||x + y||2 + ||x - y||2 = 2 ||x||2 + 2||y||2
Show that C(X) (exercise 2.85) is not an inner product space.Two vectors x and y in an inner product space X are orthogonal if xT y = 0. We symbolize this by x Š¥ y The orthogonal complement
Any pairwise orthogonal set of nonzero vectors is linearly independent.
Let the matrix A = (aij) represent a linear operator with respect to an orthonormal basis x1; x2,..., xn for an inner product space X. Then aij = xTf (xj) for every i, j A link between the inner
For any two nonzero elements x and y in an inner product space X, define the angle y between x and y byfor 0 ‰¤ Î¸ ‰¤ n. Show that 1. - 1 ‰¤ cos
If x ⊥ y, then ||x + y||2 = ||x||2 + ||y||2 The next result provides the crucial step in establishing the separating hyperplane theorem (section 3.9).
Let S be a nonempty, closed, convex set in a Euclidean space X and y a point outside S (figure 3.4). Show that 1. There exists a point x0 e S which is closest to y, that is, ||x0 - y|| ≤ ||x - y||
Generalize the preceding exercise to any Hilbert space. Specifically, let S be a nonempty, closed, convex set in Hilbert space X and y B S. LetThen there exists a sequence (xn) in S such that || xn -
Let S be a closed convex subset of a Euclidean space X and T be another set containing S. There exists a continuous function g: T → S that retracts T onto S, that is, for which g(x) = x for every x
Let f ∊ X* be a continuous linear functional on a Hilbert space X. There exists a unique element y ∊ X such that f(x) = xT y for every x ∊ X
If X is a Hilbert space, then so is X*.
Every Hilbert space is reflexive.
Let f ∊ L(X, Y) be a linear function between Hilbert spaces X and Y. 1. There exists a unique x* ∊ X such that fy(x) = xT x*. 2. Define f*: Y → X by f*(y) = x*. Then f * satisfies f (x)T y = xT
Verify that the Shapley value is a feasible allocation, that is,This condition is sometimes called Pareto optimality in the literature of game theory.
Let A, B, and C be matrices that differ only in their ith row, with the ith row of C being a linear combination of the rows of A and B. That is,Then det(C) = Î± det(A) + Î²
Show that the eigenvectors corresponding to a particular eigen-value, together with the zero vector 0X, form a subspace of X
A linear operator is singular if and only if it has a zero eigenvalue.
Let f be a linear operator on a Euclidean space, and let the matrix A = (aij) represent f with respect to an orthonormal basis. Then f is a symmetric operator if and only if A is a symmetric matrix,
For a symmetric operator, the eigenvectors corresponding to distinct eigenvalues are orthogonal.
Let f be a symmetric operator on a Euclidean space X. Let S be the unit sphere in X, that is S = {x ∊ X: ||x|| = 1}, and define g: X × X → ℜ by g(x, y) = (
Let S be defined as in the preceding proof. Show that 1. S is a subspace of dimension n - 1 2. f(S) ⊆ S
The determinant of symmetric operator is equal to the product of its eigenvalues.
Two players i and j are substitutes in a game (N, w) if their contributions to all coalitions are identical, that is, if W(S ∪ {i}) = w(S ∪ {j} for every S ⊆ N \ {i, j} Verify that the Shapley
Let the matrix A = (aij) represent a linear operator f with respect to the orthonormal basis x1; x2; . . . ; xn. Then the sumdefines a quadratic form on X, where x1; x2; . . . ; xn are the
For any quadratic form Q(x). xTAx, there exists a basis x1; x2; . . . , xn and numbers such that Q(x).
1. Show that the quadratic form (11) can be rewritten asassuming that a11 ‰ 0. This procedure is known as ``completing the square.'' 2. Deduce (12). 3. Deduce (13). This is an example of
Show that Q(0) . 0 for every quadratic form Q. Since every quadratic form passes through the origin (exercise 3.93), a positive definite quadratic form has a unique minimum (at 0). Similarly a
A positive (negative) definite matrix is nonsingular
A positive definite matrix A = (aij) has a positive diagonal, that is, A positive definite ⇒ aii > 0 for every i One of the important uses of eigenvalues is to characterize definite matrices, as
A symmetric matrix is
A nonnegative definite matrix A is positive definite if and only if it is nonsingular.
Verify these assertions directly.
Show that the function f(x) = 10x - x2 represents the total revenue function for a monopolist facing the market demand curve x = 10 - p where x is the quantity demanded and p is the market price. In
Let f: X †’ Y be differentiable at x0 with derivative Df[x0], and let x be a vector of unit norm (||x|| = 1). Show that the directional derivative of f at x0 in the direction x is the value
Show that the ith partial derivative of the function f: „œn †’ „œ at some point x0 corresponds to the directional derivative of f at x0 in the direction ei, whereei =
Calculate the directional derivative of the functionat the point (8, 8) in the direction (1, 1).
Show that the gradient of a differentiable functional on „œn comprises the vector of its partial derivatives, that is,
Show that the derivative of a functional on „œn can be expressed as the inner product
If a differentiable functional f is increasing, then ∇f(x) ≥ 0 for every x ∈ X; that is, every partial derivative Dxi f[x] is nonnegative.
Show that the gradient of a differentiable function f points in the direction of greatest increase.
f: X → ℜm, X ⊂ ℜn is differentiable x0 if and only if each component fj is differentiable at x0. The matrix representing the derivative, the Jacobian, comprises the partial derivatives of the
A point x0 is a regular point of a C1 operator if and only det Jf (x0) ≠ 0.
Suppose that nominal GDP rose 10 percent in your country last year, while prices rose 5 percent. What was the growth rate of real GDP?
A point x0 is a critical point of a C1 functional if and only if ∇
Every continuous bilinear function f: X × Y → Z is differentiable with Df[x, y] = f(x, ∙) + f(∙, y) that is, Df[x0, y0](x, y) = f(x0, y) + f(x, y0)
Let f: X → ℜ be differentiable at x and g: Y → ℜ be differentiable at y. Then their product f g: X × Y → R is differentiable at (x, y) with derivative Dfg[x, y] = f(x)Dg[y] + g(y) Df[x]
The power function (example 2.2) f(x) = xn, n = 1, 2, . . . is differentiable with derivative Df[x] = f′[x] = nxn-1
Assume that the inverse demand function for some good is given by p = f(x) where x is the quantity sold. Total revenue is given by R(x) = f(x)x Find the marginal revenue at x0.
Suppose that f: X †’ Y is differentiable at x and its derivative is nonsingular. Suppose further that f has an inverse f-1: Y †’ X that is continuous (i.e., f is a homeomorphism).
When the roles are reversed in the general power function, we have the general exponential function defined as f(x) = ax where a ∈ ℜ+. Show that the general exponential function is differentiable
Let f: X †’ „œ be differentiable at x where f(x) ‰ 0, then 1/f is differentiable with derivative
Show that the definition (2) can be equivalently expressed as
Let f: X †’ „œ be differentiable at x and g: Y †’ „œ be differentiable at y with f(y) ‰ 0. Then their quotient f/g: X Ã— Y †’
Calculate the value of the partial derivatives of the function f(k, l) = k2/3 l1/3 at the point (8, 8)
Show that gradient of the Cobb-Douglas functioncan be expressed as
Compute the partial derivatives of the CES function (exercise 2.35)
Suppose that f ∈ C[a, b] is differentiable on the open interval (a, b). Then there exists some x ∈ (a, b) such that f(b) - f(a) = f′[x](b - a)
A differentiable functional f on a convex set S ⊆ ℜn is increasing if and only ∇f(x) ≥ 0 for every x ∈ S, that is, if every partial derivative Dxi f[x] is nonnegative.
A differentiable functional f on a convex set S ⊆ ℜn is strictly increasing if ∇f(x) > 0 for every x ∈ X, that is, if every partial derivative Dxi f[x] is positive.
Let f be a functional on an open subset S of „œn. Then f is continuously differentiable (C1) if and only if each of the partial derivatives Dif[x] exists and is continuous on S.We now
A differentiable function f on a convex set S is constant if and only if Df[x] = 0 for every x ∈ S.
Let fn: S †’ Y be a sequence of C1 functions on an open set S, and defineSuppose that the sequence of derivatives Dfn converges uniformly to a function g: S †’ BL(X, Y). Then f is
The derivative of a function is unique.
Prove that ex+y = exey for every x, y ∈ ℜ.
The elasticity of a function f: „œ †’ „œ is defined to beIn general, the elasticity varies with x. Show that the elasticity of a function is constant if and only if it
Let f: S †’ Y be a differentiable function on an open convex set S Š† X. For every x0, x1, x2 ˆˆ S,
Let f: X → Y be C1. For every x0 ∈ X and
Let f: X → Y be C1. For every x0 ∈ X and Discuss.

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